I. Field of the Invention
The present invention relates to cellular communications and also relates to the Nyquist rate for data symbol transmission, the Shannon bound on communications capacity, and symbol modulation and demodulation for high-data-rate satellite, airborne, wired, wireless, and optical communications and includes all of the communications symbol modulations and the future modulations for single links and multiple access links which include electrical and optical, wired, mobile, point-to-point, point-to-multipoint, multipoint-to-multipoint, cellular, multiple-input multiple-output MIMO, terrestrial networks, and satellite communication networks. In particular it relates to WiFi, WiMax and long-term evolution LTE for cellular communications and satellite communications. WiFi, WiMax use orthogonal frequency division multiplexing OFDM on both links and LTE uses single carrier OFDM (SC-OFDM) on the uplink from user to base station and OFDM on the downlink form base station to user. WiMax occupies a larger frequency band than WiFi and both use OFDM waveforms. SC-OFDM LTE is a single carrier orthogonal waveform version of OFDM which uses orthogonal frequency subbands of varying widths.
II. Description of the Related Art
Bounds on current communications capacity are the communications Nyquist rate and the Shannon capacity theorem. The Nyquist rate is the complex digital sampling rate equal to B that is sufficient to include all of the information within a frequency band B over a communications link. For communications, equivalent expressions for the Nyquist rate bound are defined in equations (1).Ts≧1/B BTs≧1  (1)wherein 1/Ts is the data symbol transmission rate in the frequency band B which means Ts is the spacing between the data symbols.
The Shannon bound for the maximum data rate C is a bound on the corresponding number of information bits per symbol b as well as a bound on the communications efficiency η and is complemented by the Shannon coding theorem, and are defined in equations (2).
Shannon Bounds and Coding Theorem
1 Shannon capacity theoremC=B log2(1+S/N)Channel capacity in bits/second=Bps for an additive white Gaussian noise AWGN channel with bandwidth B wherein “log2” is the logarithm to the base 2=Maximum rate at which information can be reliably transmitted over a noisy channel where S/N is the signal-to-noise ratio in B  (2)
2 Shannon bound on b, η, and Eb/No
                              max          ⁢                      {            b            }                          =                ⁢                  max          ⁢                      {                          C              /              B                        }                                                  =                ⁢                              log            2                    ⁡                      (                          1              +                              S                /                N                                      )                                                  =                ⁢                  max          ⁢                      {            η            }                              Eb/No=[2^max{b}−1]/max{b}                wherein                    b=CB in Bps/Hz=Bits/symbol            η=b/TsB, Bps/Hz            Ts=symbol interval                        
3 Shannon coding theorem for the information bit rate Rb                 For Rb<C there exists codes which support reliable communications        For Rb>C there are no codes which support reliable communicationswherein Eb/No is the ratio of energy per information bit Eb to the noise power density No, max{b} is the maximum value of the number of information bits per symbol b and also is the information rate in Bps/Hz, and since the communications efficiency η=b/(TsB) in bits/sec/Hz it follows that maximum values of b and η are equal. Derivation of the equation for Eb/No uses the definition Eb/No=(S/N)/b in addition to 1 and 2. Reliable communications in the statement of the Shannon coding theorem 3 means an arbitrarily low bit error rate BER.        
MIMO communications enable higher capacities to be supported with multiple independent links over the same bandwidth. This multiple-input multiple-output MIMO requires the physical existence of un-correlated multiple communications paths between a transmitter and a receiver. MIMO uses these multiple paths for independent transmissions when the transmission matrix specifying these paths has a rank and determinant sufficiently large to support the paths being used. In MIMO U.S. Pat. No. 7,680,211 a method is disclosed for constructing architectures for multiple input transmit and multiple output receive MIMO systems with generalized orthogonal space-time codes (C0) which are generalization of space-time codes C and generalizations (H0) of the transmission matrix (H) that enable the MIMO equation Y=Hf(C,X)+No to be written Y=H0C0X+No which factors out the input signal symbol vector X and allows a direct maximum-likelihood ML calculation of the estimate {circumflex over (X)} of X, and wherein Y is the received symbol vector, No is the received noise vector, and f(C,X) is a non-separable encoding C of X
FIG. 1 defines the orthogonal frequency division multiplexing OFDM waveform for the WiFi 802.16 standard power spectrum 1, 2 which implements the inverse FFT (IFFT=FFT−1) to generate OFDM (or equivalently OFDMA which is orthogonal frequency division multiple access to emphasize the multiple access applications) data symbol tones 2 over the first 3.2 μs of the 4 μs data packet in 30 in FIG. 7 with some rolloff of the tones at their ends for spectral containment. Data symbol tones are modulated with 4PSK, 16QAM, 64QAM, 256QAM depending on the transmission range and data rate and for 256QAM using the code rate option R=¾ yields the information rate b=6 Bps/Hz for the WiFi standard, with other code options available. The N=64 point FFT−1 generates N=64 tones in 2 over the 20 MHz WiFi band with 48 tones used for data transmission. In 3 the WiFi parameters are defined including a calculation of the maximum data rate Rb=57 Mbps. Later versions of WiFi allow WiFi bands of 1.25, 5, 10, 20 MHz corresponding to N=4, 16, 32, 64. For this representative OFDM WiFi QLM disclosure we are considering the WiFi standard in FIG. 1. The maximum data rate supported by WiFi standard is calculated in 3 to be ˜57 Mbps using 256QAM modulation and wherein “˜” represents an approximate value. OFDM uses pulse waveforms in time and relies on the OFDM tone modulation to provide orthogonality. SC-OFDM is a pulse-shaped OFDM that uses shaped waveforms in time to roll-off the spectrum of the waveform between adjacent channels to provide orthogonality, allows the user to occupy subbands of differing widths, and uses a different tone spacing, data packet length, and sub-frame length compared to OFDM for WiFi, WiMax. In addition to these applications the symbol modulations 4PSK, 16QAM, 64QAM, 256QAM are used for satellite, terrestrial, optical, and all other communication links and with maximum data symbol rates achieved using 256QAM.